EXERCISE: THE DERIVATIVES
1.
2.
3.
Find the following limits: 4x 2 − x a) lim x →− 2 x 3 − 5
b)
lim
1 x→ 2
6x − 3 x(1 − 2x )
2 x + 2 ; x −2 Given a piecewise function: f ( x ) = x3 − 4 ; − 2 x 2 4 x −1 ; x 2 a) Find the limits of f(x) at x = –2, x = 2 and x approaches ∞. b) Determine the continuity of f(x) at x = 2. Determine the continuity of g(x) at x = –5 and x = –3 given x + 3 , − 5 x −3 g ( x ) = 0 , x = −3 2 9 − x , x −3
4.
By using the first principles, find the differentiation of the function: f ( x ) = x −
5.
Find dy/dx: a)
y = 2x + e − 32x + tan−1 2x
b)
y = ( 6 − 3x ) (1 + x )
c)
3x 2 y = sin + x 7x + 5
d)
y=
e)
y = ln (ln2x )
f)
2
2
x2 − x x2 + 1
3y3 x + ( x + y ) = 3 − 2x 3
ALL RIGHTS RESERVED MOOC MAT099
1
EXERCISE: THE DERIVATIVES
ANSWERS:
1.
2.
a)
0
b)
–6
a) b)
3.
lim f ( x ) does not exist ; lim f ( x ) = 4 ; lim f ( x ) = 0
x →−2
x→2
x →
f ( x ) continuous at x = 2
g ( x ) discontinuous at x = −5 g ( x ) continuous at x = −3
4.
f ' ( x) =
5.
a) b) c)
1
2 x− dy 1 2 = − 32x ( 2ln3 ) + dx 2x + e 1 + 4x2 dy = 2 6 + 3x − 3x 2 ( 3 − 6x ) dx dy 2 15 2 = − 2 cos + dx x x ( 7x + 5 )2
(
)
d)
dy x 2 + 2x − 1 = 2 dx x2 + 1
e)
dy 1 = dx x ln 2x
f)
3 dy −2 − 3y − 3 ( x + y ) = 2 dx 9xy 2 + 3 ( x + y )
(
)
2
ALL RIGHTS RESERVED MOOC MAT099
2